Non-planar corrections in ABJM theory from quantum M2 branes

The quantization of semiclassical strings in AdS spacetimes yields predictions for the strong-coupling behaviour of the scaling dimensions of the corresponding operators in the planar limit of the dual gauge theory. Finding non-planar corrections requires computing string loops (corresponding to torus and higher genus surfaces), which is a challenging task. It turns out that in the case of the $U_k(N) \times U_{-k}(N)$ ABJM theory there is an alternative approach: one may semiclassically quantize M2 branes in AdS$_{4}\times S^7/\mathbb{Z}_{k}$ which are wrapped around the 11d circle of radius $1/k= \lambda/N$. Such M2 branes are the M-theory generalization of the strings in AdS$_4\times$CP$^3$. In this work, we show that by expanding in large M2 brane tension $T_2 \sim \sqrt{kN}$ for fixed $k$, followed by an expansion in large $k$, we can predict the large $\lambda$ asymptotics of the non-planar corrections to the dimensions of the dual ABJM operators. As a specific example, we consider the M2 brane configuration that generalizes the long folded string with large spin in AdS$_4$, and compute the 1-loop correction to its energy. This calculation allows us to determine non-planar corrections to the universal scaling function or cusp anomalous dimension. We extend our analysis to the semiclassical M2 branes that generalize the "short" and "long" circular strings with two equal angular momenta in CP$^3$. The "short" M2 brane corresponds to a dual operator whose dimension at strong coupling scales as $\Delta \sim \lambda^{1/4} + \dots$, and we derive the leading non-planar correction to it.